chapter4.Rmd
Valeria Caras
11/20/2019
Exercise 4
## Data overview
library(MASS)
data("Boston")
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00#variables distribution
p <- pairs(Boston)
p## NULL#calculate the correlation matrix and round it
cor_matrix<-cor(Boston)
#print the correlation matrix
cor_matrix## crim zn indus chas nox
## crim 1.00000000 -0.20046922 0.40658341 -0.055891582 0.42097171
## zn -0.20046922 1.00000000 -0.53382819 -0.042696719 -0.51660371
## indus 0.40658341 -0.53382819 1.00000000 0.062938027 0.76365145
## chas -0.05589158 -0.04269672 0.06293803 1.000000000 0.09120281
## nox 0.42097171 -0.51660371 0.76365145 0.091202807 1.00000000
## rm -0.21924670 0.31199059 -0.39167585 0.091251225 -0.30218819
## age 0.35273425 -0.56953734 0.64477851 0.086517774 0.73147010
## dis -0.37967009 0.66440822 -0.70802699 -0.099175780 -0.76923011
## rad 0.62550515 -0.31194783 0.59512927 -0.007368241 0.61144056
## tax 0.58276431 -0.31456332 0.72076018 -0.035586518 0.66802320
## ptratio 0.28994558 -0.39167855 0.38324756 -0.121515174 0.18893268
## black -0.38506394 0.17552032 -0.35697654 0.048788485 -0.38005064
## lstat 0.45562148 -0.41299457 0.60379972 -0.053929298 0.59087892
## medv -0.38830461 0.36044534 -0.48372516 0.175260177 -0.42732077
## rm age dis rad tax
## crim -0.21924670 0.35273425 -0.37967009 0.625505145 0.58276431
## zn 0.31199059 -0.56953734 0.66440822 -0.311947826 -0.31456332
## indus -0.39167585 0.64477851 -0.70802699 0.595129275 0.72076018
## chas 0.09125123 0.08651777 -0.09917578 -0.007368241 -0.03558652
## nox -0.30218819 0.73147010 -0.76923011 0.611440563 0.66802320
## rm 1.00000000 -0.24026493 0.20524621 -0.209846668 -0.29204783
## age -0.24026493 1.00000000 -0.74788054 0.456022452 0.50645559
## dis 0.20524621 -0.74788054 1.00000000 -0.494587930 -0.53443158
## rad -0.20984667 0.45602245 -0.49458793 1.000000000 0.91022819
## tax -0.29204783 0.50645559 -0.53443158 0.910228189 1.00000000
## ptratio -0.35550149 0.26151501 -0.23247054 0.464741179 0.46085304
## black 0.12806864 -0.27353398 0.29151167 -0.444412816 -0.44180801
## lstat -0.61380827 0.60233853 -0.49699583 0.488676335 0.54399341
## medv 0.69535995 -0.37695457 0.24992873 -0.381626231 -0.46853593
## ptratio black lstat medv
## crim 0.2899456 -0.38506394 0.4556215 -0.3883046
## zn -0.3916785 0.17552032 -0.4129946 0.3604453
## indus 0.3832476 -0.35697654 0.6037997 -0.4837252
## chas -0.1215152 0.04878848 -0.0539293 0.1752602
## nox 0.1889327 -0.38005064 0.5908789 -0.4273208
## rm -0.3555015 0.12806864 -0.6138083 0.6953599
## age 0.2615150 -0.27353398 0.6023385 -0.3769546
## dis -0.2324705 0.29151167 -0.4969958 0.2499287
## rad 0.4647412 -0.44441282 0.4886763 -0.3816262
## tax 0.4608530 -0.44180801 0.5439934 -0.4685359
## ptratio 1.0000000 -0.17738330 0.3740443 -0.5077867
## black -0.1773833 1.00000000 -0.3660869 0.3334608
## lstat 0.3740443 -0.36608690 1.0000000 -0.7376627
## medv -0.5077867 0.33346082 -0.7376627 1.0000000#visualize the correlation matrix
library(corrplot)## corrplot 0.84 loadedlibrary(tidyverse)## ── Attaching packages ───────────────────────────────── tidyverse 1.2.1 ──## ✔ ggplot2 3.2.1 ✔ purrr 0.3.3
## ✔ tibble 2.1.3 ✔ dplyr 0.8.3
## ✔ tidyr 1.0.0 ✔ stringr 1.4.0
## ✔ readr 1.3.1 ✔ forcats 0.4.0## ── Conflicts ──────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ✖ dplyr::select() masks MASS::select()
corrplot(cor_matrix, method="circle")
corrplot(cor_matrix, method="circle", type = "upper",cl.pos = "b", tl.pos = "d",tl.cex = 0.6)%>%round()
## crim zn indus chas nox rm age dis rad tax ptratio black lstat medv
## crim 1 0 0 0 0 0 0 0 1 1 0 0 0 0
## zn 0 1 -1 0 -1 0 -1 1 0 0 0 0 0 0
## indus 0 -1 1 0 1 0 1 -1 1 1 0 0 1 0
## chas 0 0 0 1 0 0 0 0 0 0 0 0 0 0
## nox 0 -1 1 0 1 0 1 -1 1 1 0 0 1 0
## rm 0 0 0 0 0 1 0 0 0 0 0 0 -1 1
## age 0 -1 1 0 1 0 1 -1 0 1 0 0 1 0
## dis 0 1 -1 0 -1 0 -1 1 0 -1 0 0 0 0
## rad 1 0 1 0 1 0 0 0 1 1 0 0 0 0
## tax 1 0 1 0 1 0 1 -1 1 1 0 0 1 0
## ptratio 0 0 0 0 0 0 0 0 0 0 1 0 0 -1
## black 0 0 0 0 0 0 0 0 0 0 0 1 0 0
## lstat 0 0 1 0 1 -1 1 0 0 1 0 0 1 -1
## medv 0 0 0 0 0 1 0 0 0 0 -1 0 -1 1This data has 506 obs. of 14 variables which contain information about different indicators of Boston city as per capita crime rate, index of accessibility to radial highways and etc. In the first plot we see the distibution of variables. It looks like only rm has a normal distribution while crime, chas, lstat are shifted to the left, age to the right and etc. In the second plot positive correlations are displayed in blue and negative correlations in red color. Color intensity and the size of the circle are proportional to the correlation coefficients. For example, the is strong positive correlation between such variables as crime and rad and tax, between indus and nox, age and tax. Negative correlations between dis and nox,age and dis, medv and lstat.
## Scaling and factor variable
#center and standardize variables
boston_scaled <- scale (Boston)
#summaries of the scaled variables
summary(boston_scaled) ## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865#class of the boston_scaled object
class(boston_scaled)## [1] "matrix"#change the object to data frame
boston_scaled <- as.data.frame(boston_scaled)
#create a quantile vector of crim and print it
bins <- quantile(boston_scaled$crim)
bins## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610#create a categorical variable 'crime'
vector <- c("low","med_low","med_high","high")
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, label=vector)
#look at the table of the new factor crime
table(crime)## crime
## low med_low med_high high
## 127 126 126 127# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)
#add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)
## Train and test sets
#number of rows in the Boston dataset
n <- nrow(boston_scaled)
#choose randomly 80% of the rows
ind <- sample(n, size = n * 0.8)
#create train set
train <- boston_scaled[ind,]
#create test set
test <- boston_scaled[-ind,]
#save the correct classes from test data
correct_classes <- c(test$crime)
#remove the crime variable from test data
test <- dplyr::select(test, -crime)
str(train)## 'data.frame': 404 obs. of 14 variables:
## $ zn : num -0.487 -0.487 -0.487 -0.487 -0.487 ...
## $ indus : num -1.2648 1.015 -0.4368 -0.0797 -0.0797 ...
## $ chas : num -0.272 3.665 -0.272 3.665 -0.272 ...
## $ nox : num -0.576 1.858 -0.144 -0.567 -0.567 ...
## $ rm : num 1.239 0.157 -0.303 -1.339 -1.242 ...
## $ age : num 0.839 0.797 1.116 1.116 -2.088 ...
## $ dis : num -0.5197 -0.6125 0.1804 0.038 -0.0986 ...
## $ rad : num -0.752 1.66 -0.637 -0.637 -0.637 ...
## $ tax : num -1.277 1.529 -0.601 -0.779 -0.779 ...
## $ ptratio: num -0.3028 0.8058 1.1753 0.0667 0.0667 ...
## $ black : num 0.4102 0.3797 0.2197 0.4406 -0.0848 ...
## $ lstat : num -1.0969 0.0864 0.0542 1.4615 2.3662 ...
## $ medv : num 1.6709 -0.0906 -0.8734 -0.2754 0.1269 ...
## $ crime : Factor w/ 4 levels "low","med_low",..: 2 4 3 3 3 4 3 2 1 1 ...str(test)## 'data.frame': 102 obs. of 13 variables:
## $ zn : num 0.2845 -0.4872 0.0487 -0.4872 -0.4872 ...
## $ indus : num -1.287 -1.306 -0.476 -0.437 -0.437 ...
## $ chas : num -0.272 -0.272 -0.272 -0.272 -0.272 ...
## $ nox : num -0.144 -0.834 -0.265 -0.144 -0.144 ...
## $ rm : num 0.413 1.015 -0.388 -0.641 -0.498 ...
## $ age : num -0.1199 -0.8091 -0.0702 -0.429 -1.3953 ...
## $ dis : num 0.14 1.077 0.838 0.334 0.334 ...
## $ rad : num -0.982 -0.752 -0.522 -0.637 -0.637 ...
## $ tax : num -0.666 -1.105 -0.577 -0.601 -0.601 ...
## $ ptratio: num -1.458 0.113 -1.504 1.175 1.175 ...
## $ black : num 0.441 0.416 0.426 0.427 0.331 ...
## $ lstat : num -1.0745 -1.3602 -0.0312 -0.5858 -0.8504 ...
## $ medv : num 0.1595 1.1816 0.0399 -0.2863 0.0617 ...Here we scale the data which is an operation when we subtract the column means from the corresponding columns and divide the difference with standard deviation. It helps us to have normal distribution of variables later used in clastering. When we create a factor variable’crim’ and use the quantiles as the break points to the variable.Later we divide the dataset to train and test sets, so that 80% of the data belongs to the train set.
# Linear discriminant analysis
lda.fit <- lda(crime ~ ., data = train)
#print the lda.fit object
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2475248 0.2450495 0.2549505 0.2524752
##
## Group means:
## zn indus chas nox rm
## low 0.95964878 -0.8673138 -0.154216061 -0.8872033 0.3852534
## med_low -0.06843018 -0.3252050 0.006051757 -0.5687651 -0.1266521
## med_high -0.40315105 0.2249195 0.186362222 0.4002722 0.1243569
## high -0.48724019 1.0171096 -0.079333958 1.0567350 -0.4911568
## age dis rad tax ptratio
## low -0.8763374 0.8941028 -0.6947544 -0.7428462 -0.44921851
## med_low -0.4039546 0.4234694 -0.5422055 -0.4700267 -0.05551182
## med_high 0.4401306 -0.3902071 -0.4065227 -0.2966375 -0.29696463
## high 0.8353906 -0.8435092 1.6382099 1.5141140 0.78087177
## black lstat medv
## low 0.38368498 -0.75302015 0.49746025
## med_low 0.31167860 -0.13033637 -0.01422036
## med_high 0.05748202 0.01581784 0.16881733
## high -0.61624939 0.90741102 -0.69933165
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.12429716 0.69753574 -1.08676661
## indus 0.03308534 -0.14148348 0.13250394
## chas -0.08749391 -0.08730002 0.17014276
## nox 0.32783864 -0.76478327 -1.05103995
## rm -0.09991114 -0.16357096 -0.09662516
## age 0.27161792 -0.33897543 -0.36267828
## dis -0.09843482 -0.22843932 0.29254261
## rad 3.19026316 1.01675964 -0.40923712
## tax -0.01480205 -0.11142125 0.88024354
## ptratio 0.12775748 -0.02921722 -0.16575052
## black -0.08861513 0.03753831 0.12940586
## lstat 0.20520684 -0.22684242 0.48875493
## medv 0.17946148 -0.39341523 -0.11656291
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9468 0.0404 0.0129#the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
#target classes as numeric
classes <- as.numeric(train$crime)
#plot the lda results
plot(lda.fit, dimen = 2,col = classes,pch = classes)
lda.arrows(lda.fit, myscale = 2)
We have 4 clasters, so train data was devided in 25% 4 times with crime as target variable. In the plot we see that three clasters are overlapping while the 4rd one in quite far for them. We aslo see that such variables as zn, nox, rad, ptratio have a big impact on the model
# Predictors
#create train set
train <- boston_scaled[ind,]
#create test set
test <- boston_scaled[-ind,]
#save the correct classes from test data
correct_classes <- c(test$crime)
#remove the crime variable from test data
test <- dplyr::select(test, -crime)
#predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## 1 15 11 1 0
## 2 4 14 9 0
## 3 0 7 15 1
## 4 0 0 0 25#target classes as numeric
classes <- as.numeric(correct_classes)
#plot the lda results
plot(lda.fit, dimen = 2,col = classes,pch = classes)
lda.arrows(lda.fit, myscale = 2)
In the table we see the relation between the correct classes and the predicted ones. The classifier did not predict the crime rates correctly since predicted numbers are higher than correct. I also tried to vizualize the LDA results for crime in test data and we see that everything is in a mess.
## distances
library(MASS)
data('Boston')
#scale the data
boston_scaled <- scale (Boston)
#euclidean distance matrix
dist_eu <- (boston_scaled)
#look at the summary of the distances
summary(dist_eu)## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865#manhattan distance matrix
dist_man <- dist(boston_scaled, method = "manhattan")
#look at the summary of the distances
summary(dist_man)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2662 8.4832 12.6090 13.5488 17.7568 48.8618# k-means clustering
km <-kmeans(boston_scaled, centers = 1)
km <-kmeans(boston_scaled, centers = 3)
km <-kmeans(boston_scaled, centers = 2)
# plot the Boston dataset with clusters
pairs(boston_scaled, col=km$cluster)
#determine K
set.seed(123)
library(ggplot2)
#determine the number of clusters
k_max <- 10
#calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled, k)$tot.withinss})
#visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')
#k-means clustering
km <-kmeans(boston_scaled, centers = 2)Based on the calculated distance meajure the k-classtering was maded. On the graph we see a significant change in point 2 - so the optimal number of clastters is 2. On the plot we see the vizualization for 2 clasters - which part of data where belongs.
## 3D plot
model_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutplot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers')